# Mathematical Software

## Projects

We develop different software packages:

OSCAR

The OSCAR project uses the Julia programming language to integrate the four computer algebra systems GAP, polymake, Singular, and Antic (Hecke, Nemo) developed within the SFB/TRR 195 into a visionary next generation open source computer algebra system for computations in algebra, geometry, and number theory surpassing the combined mathematical capabilities of the underlying systems. Some of the chair's scientific staff are actively involved in the development.

jets

The Maple library jets was originally meant to be an extension of the package desolv, where functions for computating generalized symmetries were added. Now it is an independent package, whose capabilities are beyond computating symmetries. Further informations can be found on the webpage of Mohamed Barakat.

Involutive and Janet

The Janet-algorithm produces a normal form for systems of linear partial differential equations respectively for systems of polynomial equations. Implementations are now available for Maple. The packages Janet and Involutive offer various other functions for concrete applications.

One of these applications ist for example the control theory.

Thomas Decomposition

The Thomas-algorithm constructs a decomposition of systems of algebric or differential equations and inequations into simple systems.

CoW

The CoW package for Maple implements the computation of the comprehensive weight enumerator of a linear code.

homalg

The homalg project is meant as a continiously growing, source open book in several volumes about homological and homotopical algebra. homalg tries to translate as much homological algebra into a language understandable for a computer as possible. But homalg also should be readable in wide parts for mathematicans without programming experience. GAP4 was choosen as language for homalg for a huge amount of reasons.

conley

An application of homalg is the package conley. This package calculates connection and stochastic matrices of Morse decompositions of a dynamical system.

PSL

The Maple package PSL is an implementation of methods for recognizing groups of type L_2 as an epimorphic image of a finite representable group.

QuillenSuslin

Quillen Suslin is a Maple package, which implements algorithms for calculating bases of free moduls over polynomial rings.

CARAT

By using the package CARAT, it is possible to construct and classificate cristallographycal space groups up to dimension 6. Further informations can be found at the CARAT homepage. The newest version is available for download there.

OreModules

*OreModules*
is a Maple implementation of algorithms which compute parametrizations, extension modules (ext), resolutions and other algebraic objects for linear systems of differential equations, time-delay systems, etc. The package *OreModules*, based on an original program by F. Chyzak and A. Quadrat, is maintained and further developed by A. Quadrat and D. Robertz.